Cleave has two quite different meanings, to cling and to cut. "cleft" and "cloven" are also interesting participles.I would be interested in finding out if there is any etymological connection between the two meanings.

Well, the Doc's explanation makes it clear they're actually two different words. But picture this: someone walks up to a tree stump with an axe. He or she swings the axe and cleaves the stump in two, only to find the axe is cleaving to the newly-split wood.